Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(9,10,15,12)(11,13,18,16)(14,17,23,20)(19,21,24,22)(25,26,30,33)(27,31,35,29)(28,36,38,32)(34,37,40,39);(1,2,5,6)(3,7,4,8)(9,11,15,18)(10,16,12,13)(14,19,23,24)(17,22,20,21);(1,3,5,4)(2,7,6,8)(10,12)(11,18)(17,20)(19,24)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(9,12,15,10)(11,13,18,16)(14,20,23,17)(19,21,24,22)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(1,3,5,4)(2,7,6,8)(9,13)(10,11)(12,18)(14,22)(15,16)(17,24)(19,20)(21,23)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(1,4,5,3)(2,8,6,7)(9,14)(10,17)(11,19)(12,20)(13,21)(15,23)(16,22)(18,24)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=gh,d^2=i,e^2=gh,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7777691);